In Villani's book, Topics in Optimal Transportation, Lemma 6.3 is claimed without proof.
Lemma 6.3 (The interpolant is supported in the Minkowski sum). Let $\mu=\rho_0$, $\nu=\rho_1$ be the uniform probability measures carried by the compact sets $X$ and $Y$. Then, for all $t\in[0,1]$, the interpolant $\rho_t=[\mu,\nu]_t$ has its support included in the Minkowski combination $(1-t)X+tY$.
Is there anyone who could tell me how to prove that?